A OTE O SPURIOUS REGRESSIO I PAELS WITH CROSS-SECTIO DEPEDECE Jen-Je Su Deparmen of Appled and Inernaonal Economcs Massey Unversy Prvae Bag - Palmerson orh ew Zealand E-mal: jjsu@masseyacnz ABSTRACT Ths paper analyses regresson of wo ndependen saonary panels wh cross-seconal dependence I s shown ha he poolng leas squares (PLS esmaor converges o zero n probably whle he ndvdual OLS esmaor converges o a random varable However, he PLS-based and he OLS-based -sascs dverge, so he null hypohess of no correlaon ends o be spurously rejeced
I ITRODUCTIO The ssue of spurous regresson s well documened n economercs; was frs suded by Granger and ewbold (974 usng smulaons and a full analycal explanaon was laer provded n Phllps (986 Spurous regressons occur when wo ndependen negraed processes are regressed on each oher I s found ha n such occasons: ( he OLS esmaor of he slope coeffcen s asympocally random, so he rue slope (zero fals o be denfed and he OLS esmaor s nconssen, and ( he -sasc of he slope does no have a lmng dsrbuon bu dverges a a T rae as he sample sze (T goes o nfny; herefore, he null hypohess of a zero slope coeffcen ends o be spurously rejeced Recenly, Kao (999 and Phllps and Moon (999 examned spurous regressons n panel daa when boh he cross-secon dmenson ( and he me-seres span (T are large For he case of regresson of wo ndependen nonsaonary panels, s found ha he poolng leas squares (PLS esmaor of he slope converges o zero n probably (so he PLS esmaor s conssen, provded ha cross-secon uns whn each panel are muually ndependen Accordng o Phllps and Moon (999, hs s because ha he srong nose effec, whch makes he slope undenfable n each ndvdual me-seres regresson, s aenuaed by he ncluson of a large amoun of ndependen cross-secon nformaon On he oher hand, he usual -sasc of he slope dverges (a a T rae, oo, mplyng ha nferences abou he slope are wrong wh he probably ha goes o one asympocally Ths noe sudes spurous regresson under a dfferen panel seng In parcular, we consder a regresson beween wo ndependen saonary panels wh cross-seconal dependence To model cross-seconal dependence n panels, we assume a facor model n each panel We esablsh he lmng dsrbuons of he PLS esmaor and he ndvdual OLS esmaor (a any gven me of he slope, and he lmng dsrbuons of he PLS-based and he OLS-based -sascs We fnd ha he PLS esmaor converges o he rue slope value (zero as n he case of regresson n cross-seconally ndependen panels (saonary or nonsaonary, bu a a dfferen convergence rae (as dscussed n Secon III On he oher hand, he OLS esmaor converges o a random varable We also fnd ha he PLS-based -sasc and he OLS-based -sasc dverge (boh a a rae and, as a resul, spurous rejecons of he zero-slope null occur The paper s organzed as follows Secon II nroduces he facor model based panels wh cross-seconal dependence Secon III derves he asympoc dsrbuons of he PLS and he PLS-based -sasc as well as he asympoc dsrbuons of he OLS esmaor and he OLSbased -sasc Secon IV concludes As a maer of noaon, hroughou he paper, ( T, denoes and T go o nfny jonly, sgnfy weak convergence, and p means convergence n probably
II CROSS-SECTIOALLY DEPEDET PAELS Le x and y, for,, and,, T, be wo ndependen panels defned as y λ f + µ, x δ g + υ Here, f and g are unobservable random facors, λ and δ are non-random facor loadng coeffcens and µ and υ are dosyncrac shocks n x and y, respecvely Smlar o Phllps and Sul (00, we assume a sngle-facor srucure o model dependence across uns See also Moon and Perron (003 and Ba and g (003 for a mul-facor panel model Followng Moon and Perron (003, we make assumpons regardng f, g, λ, δ, µ and υ as follows Assumpon m (a f θ j 0 jξ j, where ξ ~ d ( 0, and j θ j 0 j < M for some m> m (b g γζ j 0 j j, where ζ ~ d ( 0, and j γ j 0 j < M for some m> (c ξ and ζ are ndependen Assumpon (a µ d j 0 jε, j, where ε, ~ d ( 0,, across and over and wh a fne fourh momen, m nf d 0 j 0 j>, and j d j 0 j < M wh dj sup dj (b υ c j 0 jη, j, where η ~ d ( 0,, across and over and wh a fne fourh momen, m nf c 0 j 0 j>, and j c j 0 j < M wh cj sup dj (c ε and η are ndependen Assumpon 3 (a ξ and ε s are ndependen (b ζ and η s are ndependen Assumpon 4 Defne σ µ, j 0 dj and σ υ, j 0cj, where σ µ, and σ υ, sgnfes he varance of µ and υ, respecvely Le ωµ lm ωµ, and συ lm συ, Assume ha σ µ and σ υ are boh well defned Assumpon 5 (a λ m λ ( 0 and δ m δ ( 0 (b λδ ( 0, λ σ, and δ σ σ λδ λ ( 0 δ ( 0 Assumpons -3 assume ha he random facors ( f, g and he dosyncrac shocks ( µ, υ are all zero-mean saonary and hey are ndependen o one anoher Under Assumpon, snce f and g are ndependen, s easy o see ha, as T, / T T f ( 0, g ω fg, where ω fg s he long-run varance of f g oe ha, under Assumpon, snce he long-run varances of f and g are well-defned and f and g are muually ndependen, he long-run (
varance of f g s well-defned Also, by Assumpons and, x and y are consruced o be ndependen o each oher The dosyncrac shocks are assumed o be ndependen across uns (Assumpon (c The exen of cross-seconal correlaon n each panel s gven by δδ jeg ( corr( x, x j, / / ( δ Eg ( + E( υ ( δ j Eg ( + E( υj λλ je( f corr( y, y j / / ( λ E( f + E( µ ( λ j E( f + E( µ j Snce Assumpon 5 does no rule ou he possbly ha λ 0 or δ 0 for some, some crosssecon uns (n each panel may be uncorrelaed wh one anoher Assumpon 4 assumes he exsence of he long-run varances of he dosyncrac shocks III SPURIOUS PAEL REGRESSIOS Consder a smple panel regresson model y x + ε,,, ;,, T ( The PLS esmaor of s gven by T x y ˆ T x, and he PLS resduals usual sasc: e y ˆ x Then, o es he null hypohess of 0, we defne he ˆ ˆ, where sˆ sˆ / T ˆ e T x T For comparson, we also consder he OLS esmaor of he slope n ( for any gven, x y ˆ x and he OLS resduals e y ˆ x And, defne he OLS-based -sasc as ˆ 3
ˆ sˆ, where sˆ / e x Theorem Le σ f E( f and σ g E( g Under Assumpons -5, we have he followng σ / ˆ λδ ω fg ( a T 0,, ( σσ δ g + σ υ ( Le ( T,, σ / λδ ω fg ( b 0, σ σ σ σ σ σ ( Remarks ( λ f µ ( δ g υ + + ˆ σ fg λδ * ( a, σδ g / / σ δ g * ( b, * Λ Λ σ f σ f g + σ g For any, le, * * * where λ λδ δ The PLS esmaor of he slope s T -conssen Ths conrass wh he well-known fac ha he PLS esmaor s T -conssen n he convenonal panel regresson ha assumes over-me saonary and cross-un ndependence Ths also conrass wh he T - conssency acheved n nonsaonary panel regresson when cross-seconal ndependence s assumed (Kao (999 and Phllps and Moon (999 On he oher hand, he me-specfc ndvdual OLS esmaor of he slope (for any s no conssen I s also worh nong ha he PLS esmaor, once correcly scaled, converges o a normal dsrbuon On he conrary, he OLS esmaor converges o a random varable ha depends on he random facors The PLS-based -sasc and he OLS-based -sasc are boh dvergen, so he spurous resuls appear Ineresngly, he dvergence rae of he PLS-based es s deermned by, he cross-secon dmenson, only Ths s oppose o he nonsaonary panel regresson case (wh cross-seconal ndependence suded n Kao (999 and Phllps and Moon (999, n whch he -sasc dverges a a rae ha depends on T, he me-seres span, only 3 There s no need o pu any resrcon on he relave growng rae beween and T o oban he jon asympoc dsrbuons of he PLS esmaor and he PLS-based -sasc In conras, he resul obaned n Phllps and Moon (999 requres he assumpon ha grows slowly han T 4
VI COCLUSIOS In hs noe, we consder a spurous regresson of wo ndependen saonary, cross-seconally correlaed panels To model cross-seconal dependence, a sngle-facor model for each panel s assumed We fnd ha he PLS esmaor converges o zero n probably so s conssen On he oher hand, he OLS esmaor converges o a random varable We also fnd ha boh he PLS-based -sasc and he OLS-based -sasc dverge and consequenly spurous resuls occur 5
References Ba, J & g, S (003 A panc aack on un roos and conegraon, Mmeo Granger, C W J & ewbold, P (974 Spurous regresson n economercs, Journal of Economercs (, -0 Kao, C (999 Spurous regresson and resdual-based ess for conegraon n panel daa, Journal of Economercs (90, -44 Moon, H R & Perron, B (003 Tesng for a un roo n panels wh dynamc facors, CLEO Dscusson Paper Unversy of Souhern Calforna Phllps, P C B (986 Undersandng spurous regressons n economercs, Journal of Economercs (33, 3-340 Phllps, P C B & Moon, H R (999 Lnear regresson lm heory for nonsaonary panel daa, Economerca (67, 057- Phllps, P C B & Sul, D (00 Dynamc panel esmaon and homogeney esng under cross secon dependence, Mmeo 6
Appendx - Proof Of Theorem Par (-(a T / ˆ 0, m ω λδ fg ( σσ δ g + συ We frs clam ha he numeraor of ˆ : ( 0, / T fg, (A T x y m λδ ω as ( T, oe ha T T T T T x y λδ f g + λ fυ + δ g µ + υ µ (A Under Assumpon, f and g are wo ndependen saonary processes, follows ha T T / / T λδ fg λδ T fg ( λδ ωfg m ω B( 0, m, (A3 λδ fg where B( s he sandard Brownan moon and ω fg s he long-run varance of f g For he second erm on he rgh hand sde of (A, we frs noe ha T E λ fυ T T E fs f λυ s λjυj s j T T E[ fs f] E λυ s λjυj s j T T E[ f f ] λ E ( υ υ, (A4 s s s ( because E( fsυ j 0for any s, and j, and E( υυ s j 0 f j Le E[ ff ] Γ f h ( h and υ E υ υ Γ h, we have ( ( h,, ( T T s T T ( f λ ( v s ( f ( υ (A4 Γ ( s λ Γ ( s ( ( Γ s Γ s (A5 oe ha, under Assumpon (a, ( ( υ υ ( υ j, j+ h j j+ h j o j 0 ( h sup ( h sup d d d d ( h, Γ Γ Γ 7
so ha, T T s ( f ( v (A5 λ Γ ( s Γ ( s ( f ( v λ T Γ ( h Γ ( h h 0 T Γ h Γ h h 0 h 0 ( f ( υ λ ( (, by he Cauchy nequaly Due o he summably condons of Assumpons (a and (b, Γ ( υ ( h 0 h < M and Γ ( f ( h 0 h < M, for some fne M (>0 And, snce λ O (, we conclude Therefore, T E λ fυ O( T Smlarly, T / / λ fυ Op( T (A6 T / / δ gµ Op( T (A7 Also, snce µ and υ are saonary, cross-seconally ndependen, and ndependen o each oher, can be shown ha T / / υµ Op ( T (A8 By (A3 and (A6 ~ (A8, he resul of (A drecly follows We nex clam ha he denomnaor of ˆ :, (A9 T p σσg+ σ T x δ υ as ( T, Wre T T T T x δ g + δ gυ + υ (A0 T T / / T Snce T δ g pσδσg, δgυ Op( T, and T υ p συ, he resul (A9 drecly follows By (A and (A9, we complee he proof of Theorem ( 8
Proof of Theorem (-(b: Wre / σλδ ω fg 0, ( σσ λ f + σµ ( σσ δ g + συ oe ha ˆ / / s T T ˆ e x T T x y T T eˆ T y ˆ x T + T T T T y ˆ T ˆ T x y x (A T Followng he proof of (A9, s easy o show ha T y p σσ λ f + σµ Also, by he / proof of par ( n Theorem, ˆ ( O T T / p T, x Op ( T and x y Op ( T Therefore, he frs erm on he rgh hand sde of (A domnaes he oher erms n he same equaon and we conclude T T eˆ p σσ λ f + σµ (A By (A and Theorem (-(a, he resul follows Proof of Theorem (-(a σ fg ˆ λδ σδ g * I s easy o show ha x y g f and x g The resul follows σ λδ σ δ Proof of Theorem (-(b / / σδ g * * Λ Smlar o he proof of (-(a, s easy o show ˆ ˆ + e y x y x σ f σ f g + σ g Λ (A3 * * * λ λδ δ By he Theorem (-(a and (A3, he resul follows 9
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